MHB Solve Equation $\sqrt[3]{a-1}+\sqrt[3]{a}+\sqrt[3]{a+1}=0$ in Real Numbers

[anemone]

Solve in real numbers the equation $\sqrt[3]{a-1}+\sqrt[3]{a}+\sqrt[3]{a+1}=0$

 

[kaliprasad]

Solve in real numbers the equation $\sqrt[3]{a-1}+\sqrt[3]{a}+\sqrt[3]{a+1}=0$

using

Spoiler

x+y+z=0=>$x^3+y^3+x^3 = 3xyz$

we get

Spoiler

$(3a)^3 = 3a(a^2-1)$
a= 0 or +/-$\sqrt(1/8)$

 

[Opalg]

using

Spoiler

x+y+z=0=>$x^3+y^3+x^3 = 3xyz$

we get

Spoiler

$(3a)^3 = 3a(a^2-1)$
a= 0 or +/-$\sqrt(1/8)$

Neat method, but I get a different answer.

[sp]From $3a = 3\sqrt[3]{a(a^2-1)}$, I get $a^3 = a(a^2-1)$, with $a=0$ the only solution.[/sp]

 

[kaliprasad]

Neat method, but I get a different answer.

[sp]From $3a = 3\sqrt[3]{a(a^2-1)}$, I get $a^3 = a(a^2-1)$, with $a=0$ the only solution.[/sp]

There was a calculation mistake in my method

 

[anemone]

Thanks to both of you for participating and yes, $a=0$ is the only answer to the problem.:)